We don't know
Just before Obama took office in 2009, the Department of Labor released a study because, as a deputy assistant secretary explained it, "The raw wage gap continues to be used in misleading ways to advance public policy agendas without fully explaining the reasons behind the gap." The study by CONSAD Research Corp. took into account women being more likely to work part-time for lower pay, leave the labor force for children or elder care, and choose work that is "family friendly" with fuller benefit packages over higher pay. The study found that, when factoring in those variables, the gap narrows to between 93 cents and 95 cents on the dollar.
Still, a study by the American Association of University Women controlled for a number of factors, including college major, occupation, age, geographical region and hours worked, and found a persistent 7 percent wage gap between men and women a year after graduating college.
So there remains a small gap after controlling for these factors. But there are other factors that differ between men and women. E.g., men commute longer distances. PDF source.
Women, particularly women with children, tend to have shorter commuting times than men which limits the range of jobs available to them. This potentially leads to the crowding of women into those jobs available locally, and in either case, depresses wages.
But neither of the studies controlled for that.
It's also worth noting that the first two studies didn't control for the same set of variables. This means that a new study could be done combining both and adding commute times. What would that study say? We don't know.
One reason why they don't do such studies is that the more variables you use, the less reliable regression analysis is.
Another issue is that as you add more variables, the precision goes down. This is because you have to allow for the case that the error is positive or negative for all variables at once. So we tend to avoid regressions with large numbers of variables. But that doesn't mean that things aren't controlled by large numbers of correlated variables.
So we don't know the answer and are unlikely to get it. It is not possible to control for all possible variables accurately. And if we limit the variables, then it is always possible that the missing variables would fix it.